6,183 research outputs found
Polynomial sequences on quadratic curves
In this paper we generalize the study of Matiyasevich on integer points over
conics, introducing the more general concept of radical points. With this
generalization we are able to solve in positive integers some Diophantine
equations, relating these solutions by means of particular linear recurrence
sequences. We point out interesting relationships between these sequences and
known sequences in OEIS. We finally show connections between these sequences
and Chebyshev and Morgan-Voyce polynomials, finding new identities
On Some Dynamical Systems in Finite Fields and Residue Rings
We use character sums to confirm several recent conjectures of V. I. Arnold
on the uniformity of distribution properties of a certain dynamical system in a
finite field. On the other hand, we show that some conjectures are wrong. We
also analyze several other conjectures of V. I. Arnold related to the orbit
length of similar dynamical systems in residue rings and outline possible ways
to prove them. We also show that some of them require further tuning
Local Inversion of maps: Black box Cryptanalysis
This paper is a short summery of results announced in a previous paper on a
new universal method for Cryptanalysis which uses a Black Box linear algebra
approach to computation of local inversion of nonlinear maps in finite fields.
It is shown that one local inverse of the map equation can be
computed by using the minimal polynomial of the sequence defined by
iterates (or recursion) with when the sequence is
periodic. This is the only solution in the periodic orbit of the map .
Further, when the degree of the minimal polynomial is of polynomial order in
number of bits of the input of (called low complexity case), the solution
can be computed in polynomial time. The method of computation only uses the
forward computations for given which is why this is called a Black
Box approach. Application of this approach is then shown for cryptanalysis of
several maps arising in cryptographic primitives. It is shown how in the low
complexity cases maps defined by block and stream ciphers can be inverted to
find the symmetric key under known plaintext attack. Then it is shown how RSA
map can be inverted to find the plaintext as well as an equivalent private key
to break the RSA algorithm without factoring the modulus. Finally it is shown
that the discrete log computation in finite field and elliptic curves can be
formulated as a local inversion problem and the low complexity cases can be
solved in polynomial time.Comment: 13 pages. Summery and comprehension of a previous paper
arxiv.org/abs/2202.06584v
A Class of Three-Weight Cyclic Codes
Cyclic codes are a subclass of linear codes and have applications in consumer
electronics, data storage systems, and communication systems as they have
efficient encoding and decoding algorithms. In this paper, a class of
three-weight cyclic codes over \gf(p) whose duals have two zeros is
presented, where is an odd prime. The weight distribution of this class of
cyclic codes is settled. Some of the cyclic codes are optimal. The duals of a
subclass of the cyclic codes are also studied and proved to be optimal.Comment: 11 Page
- …